Complementarity in atomic (finite-level quantum) systems: an information-theoretic approach

TL;DR

This paper develops an information-theoretic framework for understanding the number-phase complementarity in atomic quantum systems, revealing new bounds and asymmetries in their uncertainty relations, and analyzing noise effects.

Contribution

It introduces a novel interpretation of number-phase complementarity using POVMs, derives tighter uncertainty bounds, and uncovers asymmetries in mutual unbiasedness in higher-dimensional systems.

Findings

Maximum knowledge of a POVM variable is less than log(dimension) bits.

Phase remains unbiased with respect to number states, but not vice versa in higher dimensions.

Noise impacts the complementarity of number and phase in single-qubit systems.

Abstract

We develop an information theoretic interpretation of the number-phase complementarity in atomic systems, where phase is treated as a continuous positive operator valued measure (POVM). The relevant uncertainty principle is obtained as an upper bound on a sum of knowledge of these two observables for the case of two-level systems. A tighter bound characterizing the uncertainty relation is obtained numerically in terms of a weighted knowledge sum involving these variables. We point out that complementarity in these systems departs from mutual unbiasededness in two signalificant ways: first, the maximum knowledge of a POVM variable is less than log(dimension) bits; second, surprisingly, for higher dimensional systems, the unbiasedness may not be mutual but unidirectional in that phase remains unbiased with respect to number states, but not vice versa. Finally, we study the effect of non-dissipative and dissipative noise on these complementary variables for a single-qubit system.